Qualitative aspects of numerical methods for integration of systems of nonlinear ordinary stochastic differential equations (SDEs) with potential applicability to mechanical engineering are presented. In particular, we study the qualitative behavior of some linearly partial-implicit midpoint-type methods for numerical integration of infinite and finite systems of SDEs with cubic-type nonlinearity and Q-regular additive space-time noise. Construction and properties such as stability and convergence of such stochastic-numerical methods is strongly related to their uniform boundedness along Lyapunov-type functionals. Well-known convergence order bounds apart from further complexity issues forces us to focus our analysis on lower order Runge-Kutta methods rather than higher order Taylor methods. Nonstandard techniques such as partial-implicit difference methods for noisy ODEs/PDEs seem to be the most promising ones in view of adequate longterm integration of such nonlinear systems.
The rate of convergence of numerical methods for integration of some convex functionals of ordinary stochastic differential equations (SDEs) is discussed. In particular, we answer how rates of p-th mean convergence carry over to rates of weak convergence for non-smooth and convex functionals of SDEs. Qualitative behavior of some numerical approximations such as nonnegativity of balanced implicit Milstein methods (BMMs) is investigated as well. Nonstandard integration techniques such as partial- and linear-implicit ones seem to be the most promissing. As a main result we obtain some justification for the choice of approximation schemes of discounted price functionals and their ingredients of random interest rates and volatility processes involved in dynamic asset pricing.
Conference Committee Involvement (1)
Noise and Fluctuations in Econophysics and Finance
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